Result 329, Functional analysis

A counterexample to metric-entropy duality

Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

How many copies of one shape are needed to cover another? The manuscript reports that a conjectured universal link between this measure of geometric complexity and the corresponding measure for polar shapes breaks down, even for cube coverings.

What changes?

The claimed counterexamples disprove Pietsch's dimension-free duality conjecture for metric entropy. A covering number counts the translated copies of one body needed to cover another; metric entropy is its logarithm. The bodies are convex and unchanged by reflection through the origin. Their polars consist of vectors whose dot product with every original point is at most one. For every proposed pair of universal constants, the manuscript constructs bodies violating the conjectured covering-entropy inequality, already when the covering body is a cube.

What does that help mathematicians do?

If established, the result rules out using the conjectured comparison to transfer covering estimates between arbitrary origin-symmetric convex bodies and their polars with constants independent of dimension. Researchers seeking such a transfer must change the comparison or impose additional restrictions. The cube case matters because choosing this familiar covering shape does not rescue the claim. Dimension-dependent comparisons and more restricted duality statements are not ruled out.

Are there practical applications?

The immediate value is foundational: it identifies a limit on how polar geometry can control covering complexity in functional analysis. Covering estimates describe how many representative points are needed to approximate a whole set. This result warns against deriving such estimates from the conjectured duality; the supplied abstract reports no practical algorithm or deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Counterexamples to the duality conjecture for metric entropy

September 24, 2026 15 pages Main result formalized in Lean

We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026,
  author = {{OpenAI}},
  title = {{Counterexamples to the duality conjecture for metric entropy}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf}{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/329.md.

A counterexample to metric-entropy duality

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Metric-entropy duality predicts a universal comparison between covering numbers of convex bodies and their polars. The formalized result disproves such a comparison: for every a,b≥1a,b\ge1, there is an origin-symmetric convex body KK in a positive finite dimension with log⁡N(K,B∞)>blog⁡N(B∞∘,a−1K∘)\log N(K,B_\infty)>b\log N(B_\infty^\circ,a^{-1}K^\circ), where NN is the least number of translates in a finite cover. A further construction makes the dual-to-primal logarithmic entropy ratio tend to zero as the dimension grows.

Comparator links

Result Comparator statement
Counterexample to metric-entropy duality MetricEntropyDuality.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.